Calculating R-Squared in Regression Analysis

The formula for calculating R-squared is:

R^2 = 1 - (SS_res / SS_tot)

Where:

- `SS_res` (Sum of Squares of Residuals) is the sum of the squared differences between the observed values and the predicted values from the model.

- `SS_tot` (Total Sum of Squares) is the sum of the squared differences between the observed values and their mean.

In formula terms:

SS_res = Σ (y_i - ŷ_i)²

SS_tot = Σ (y_i - ȳ)²

Where:

- `y_i` is an observed value.

- `ŷ_i` is a predicted value from the model.

- `ȳ` is the mean of the observed values.

By substituting these values into the formula, you calculate R-squared, which indicates how well the model explains the variability in the data.

Breaking Down RSS, TSS, and ESS for Better Regression Understanding

Understanding RSS, TSS, ESS & R² in Regression

In regression analysis, we measure how well a model explains variation in data using three core quantities: Total Sum of Squares (TSS), Residual Sum of Squares (RSS), and Explained Sum of Squares (ESS).

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🎯 Learning Goal

Understand how total variation in data is decomposed into explained and unexplained parts.

💡 Core Identity: TSS = ESS + RSS

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📘 Key Definitions

1️⃣ Total Sum of Squares (TSS)

Definition: Measures total variation in y around its mean.

TSS = Σ (y_i - y_mean)^2

• y_i → actual values
• y_mean → mean of y

💡 TSS represents total variability before modeling.

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2️⃣ Residual Sum of Squares (RSS)

Definition: Measures unexplained variation (model error).

RSS = Σ (y_i - y_hat_i)^2

• y_hat_i → predicted values

💡 RSS measures how wrong the model is.

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3️⃣ Explained Sum of Squares (ESS)

Definition: Measures variation explained by the model.

ESS = Σ (y_hat_i - y_mean)^2

💡 ESS measures how much the model explains.

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🔗 The Fundamental Relationship

TSS = ESS + RSS

The total variability in y is split into:

• Explained part (ESS)
• Unexplained part (RSS)

💡 Regression decomposes total variation into explained and residual components.

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📊 Visual Interpretation (Conceptual)

Think of It Geometrically

TSS → Distance from actual points to the mean ESS → Distance from predictions to the mean RSS → Distance from actual points to predictions

Graphically:

• Mean line → baseline model
• Regression line → improved model
• Vertical gaps → residuals

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📈 Coefficient of Determination (R²)

R^2 = ESS / TSS
R^2 = 1 - (RSS / TSS)

Interpretation

• R² = 1 → Perfect fit
• R² = 0 → No improvement over mean

💡 R² measures the proportion of total variance explained by the model.

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🧪 Step-by-Step Example Logic

How You Compute in Practice

1. Compute y_mean
2. Calculate TSS using actual values
3. Fit regression → obtain y_hat
4. Calculate RSS
5. Compute ESS = TSS − RSS
6. Compute R²

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📌 Final Summary

• TSS → Total variability
• ESS → Explained variability
• RSS → Unexplained variability

💡 A good regression model minimizes RSS and maximizes ESS.

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End of Interactive Learning Guide

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